Inverse and Joint Variation

Inverse variation

If a tank full of water is emptied in 24 minutes through a "smooth" outlet in which the area of the opening, A, is 1 square inch, an outlet in which A is 2 square inches would empty the tank twice as quickly, or in 12 minutes. And an outlet in which A is 3 square inches would empty the tank in 8 minutes.

Suppose it possible to increase or decrease A at will. We then have in t, the time required to empty the tank, and in A, the area of the opening, two related variables such that if A increases, t will decrease proportionally; while if A decreases, t will increase proportionally. That is, t and A are inversely proportional. When A is doubled, t is halved; when A is trebled, t is divided by 3; and so on.

Now let t_1 and t_2 be any two times corresponding to the areas A_1 and A_2 respectively; then

The letters in (1) say: the first time is to the second time as the second area is to the first area. In (4) the subscripts on the t's and those on the A's come in the same order, and in the form of a variation (4) becomes t \propto \dfrac1A.

In general x varies inversely as y when x varies as the reciprocal of y; that is,

x \propto \frac1y.

And if x and y denote any corresponding values of the variables, and x_1 and y_1 a particular pair of corresponding values,

x : x_1 = \frac1y : \frac{1}{y_1}, \qquad\text{whence}\qquad xy = x_1y_1.

But x_1y_1 is a constant, being the product of two definite numbers. Call this constant K; then

xy = K.

That is, if one variable varies inversely as another, the product of the two is a constant.

Example. If x varies inversely as y, and x = 6 when y is 7, find x when y is 21.

The variation is inverse. Hence x_1 : x_2 = \dfrac{1}{y_1} : \dfrac{1}{y_2}. Substituting, 6 : x_2 = \frac17 : \frac{1}{21}. Solving, x_2 = 2. (Or: xy = 42, so x = 42 \div 21 = 2.)

Joint variation

If the base of a triangle remains constant while the altitude varies, the area will vary as the altitude. Similarly, if the base varies while the altitude remains constant, the area will vary as the base. If both base and altitude vary, the area varies as the product of the two; that is, the area of the triangle varies jointly as the base and altitude. If A_1, h_1, b_1 and A_2, h_2, b_2 are the area, altitude, and base at any two times, then A_1 : A_2 = h_1b_1 : h_2b_2. In the form of a variation this becomes A \propto hb.

In general, any variable x varies jointly as two others, y and z, if

x \propto yz.

If x, y, and z denote any corresponding values of the variables, while x_1, y_1, and z_1 denote a particular set of such values, then

\frac{x}{x_1} = \frac{yz}{y_1z_1}, \qquad\text{so}\qquad x = \left(\frac{x_1}{y_1z_1}\right)yz.

But the fraction \dfrac{x_1}{y_1z_1} is a constant. Calling this constant K, we may write x \propto yz as the equation

x = Kyz.

One variable may vary directly as one variable (or several variables) and inversely as another (or several others). Also one variable may vary as the square, or the cube, or the square root, or the reciprocal, or as any algebraic expression whatever involving the other variable (or variables).

Example. If x varies jointly as y and z, and x = 24 when y is 6 and z = 8, find x when y is 9 and z = 4.

The variation is joint. Therefore \dfrac{x_1}{x_2} = \dfrac{y_1z_1}{y_2z_2}. Substituting, \dfrac{24}{x_2} = \dfrac{6 \cdot 8}{9 \cdot 4}. Solving, x_2 = 18.

Answers that contain letters are typed as usual: k m/h. Fractions and mixed numbers such as 16/3 or 5 1/3 are both fine.